Question 11 – 3

1. Question 11 – 3

2. Question 11 – 9

3. Question 11 – 9 cont.

4. Question 11 – 11 a.

5. Question 11 – 11 b, c.

6. Queueing Theory: Part II

7. Elementary Queueing Process Served customers Queueing system Queue C C C C S S Service S facility S Customers C C C C C C C Served customers

8. Relationships between and Assume that is a constant for all n. In a steady-state queueing process, Assume that the mean service time is a constant, for all It follows that,

9. The Birth-and-Death Process Most elementary queueing models assume that the inputs and outputs of the queueing system occur according to the birth-and-death process. In the context of queueing theory, the term birth refers to the arrival of a new customer into the queueing system, and death refers to the departure of a served customer.

10. The birth-and-death process is a special type of continuous time Markov chain. State: 0 1 2 3 n-2 n-1 n n+1 and are mean rates.

11. Rate In = Rate Out Principle. For any state of the system n (n = 0,1,2,…), average entering rate = average leaving rate. The equation expressing this principle is called the balance equation for state n.

12. Rate In = Rate Out State 0 1 2 n – 1 n

13. State: 0: 1: 2: To simplify notation, let for n = 1,2,…

14. and then define for n = 0. Thus, the steady-state probabilities are for n = 0,1,2,… The requirement that so that implies that

15. The definitions of L and specify that is the average arrival rate. is the mean arrival rate while the system is in state n. is the proportion of time for state n,

16. The Finite Queue Variation of the M/M/s Model] (Called the M/M/s/K Model) Queueing systems sometimes have a finite queue; i.e., the number of customers in the system is not permitted to exceed some specified number. Any customer that arrives while the queue is “full” is refused entry into the system and so leaves forever.

17. From the viewpoint of the birth-and-death process, the mean input rate into the system becomes zero at these times. The one modification is needed for n = 0, 1, 2,…, K-1 for n K. Because for some values of n, a queueing system that fits this model always will eventually reach a steady-state condition, even when

18. Question 1 Consider a birth-and-death process with just three attainable states (0,1, and 2), for which the steady-state probabilities are P0, P1, and P2, respectively. The birth-and-death rates are summarized in the following table: State Birth Rate Death Rate 0 1 2 1 1 0 _ 2 2

19. Construct the rate diagram for this birth-and-death process. • Develop the balance equations. • Solve these equations to find P0,P1, and P2. • Use the general formulas for the birth-and-death process to calculate P0 ,P1 , and P2. Also calculate L, Lq, W, and Wq.

20. Question 1 - SOLUTINON Single Serve & Finite Queue (a) Birth-and-death process 0 1 2 (b) In Out (1) Balance Equation (2) (3) (4)

21. (b)

22. From (4) so

24. Question 2 • Consider the birth-and-death process with the following mean rates. The birth rates are =2, =3, =2, =1, and =0 for n>3. The death rates are =3 =4 =1 • =2 for n>4. • Construct the rate diagram for this birth-and-death process. • Develop the balance equations. • Solve these equations to find the steady-state probability distribution P0 ,P1, ….. • Use the general formulas for the birth-and-death process to calculate P0 ,P1, ….. Also calculate L ,Lq, W, and Wq.

25. Question 2 - SOLUTION (a) 3 4 0 1 2 (b) (1) (2) (3) (4) (5) (6)

26. (c)

27. So,

28. So,

29. (d)